Asymptotic-spacing conjecture for zeros of extremal functions

Assume the extremal problem has a unique solution $\varphi_p$ for each $p>0$, and let $t_n=t_n(p)$, $n\geq1$, be the positive zeros of $\varphi_p$. Zero-spacing conjecture.

$$\lim_{n\to\infty}(t_{n+1}-t_n)=1$$

for all $p>0$. The source gives evidence in the form of an upper bound on the limsup when $p=1$.